Quests and hearts tend to do similar things: tell you want to kill, fiddle with, or retrieve from a designated area. In this regard, quests and hearts are the same. However hearts have advantages in terms of presentation.

First off, hearts are active the moment you're able to participate. If I am shooting wasps it means that I'm shooting wasps for the heart. Contrast that with a quest where you might kill a dozen bears on your way in, only to be told to go kill a dozen bears. I'd not mind that if they actually wanted two dozen, which is really just a matter of presentation: count number of bears killed before quest, add that number to the total required, thereby creating the illusion that the player got credit for them. However they don't do that. The overall effect of the hearts in this regard is to reduce the sense that I am killing mobs pointlessly. Neither system deals with post-quest/heart slaughter.

Second, hearts are simultaneous and comprehensive. I'm helping Farmer John with his farm, which includes everything from killing wasps to digging up large and aggressive grubs. Maybe all I did was kill wasps, but if I killed a ton of wasps, that's helpful, even if I ignored the grubs and leaky pipes. This even helps with the problem that a dozen other people are already there killing wasps. Since we're rendering a sort of general help, rather than heroically saving a very tiny part of the world, it makes sense that more than one person would be involved. Furthermore, having many people helps average things out, so I can imagine that despite my obsession with shooting wasps, someone did eventually get around to fixing the pipes. Quests can use the trick of stacking, having a few quests that relate to the same area, but this doesn't give the flexibility of wasp-killing vs. pipe-patching. When the quests come from multiple NPCs, particularly NPCs who aren't standing right next to each other and presumably overhearing what the other ones told me to do, then it can feel artificial, excessively planned that everyone just so happens to want me to kill a dozen bears, harvest a dozen bear asses, and kill a dozen angry stalks of grass which feed on bear corpses.